Theorem int0 4918

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4452	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3445	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4910	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 231	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 264	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2734	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3441  ∅c0 4286  ∩ cint 4903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3443  df-dif 3905  df-nul 4287  df-int 4904

This theorem is referenced by:  unissint  4928  uniintsn  4941  rint0  4944  intex  5290  intnex  5291  oev2  8452  fiint  9231  elfi2  9321  fi0  9327  cardmin2  9915  00lsp  20936  cmpfi  23356  ptbasfi  23529  fbssint  23786  fclscmp  23978  zarcmplem  34019  rankeq1o  36346  bj-0int  37277  heibor1lem  37981  ipoglb0  49275  mreclat  49278